A quantitative synthesis theory is presented for the Self-Oscillating-Adaptive-System (SOAS), whose nonlinear element has a static, odd character with hard saturation. The synthesis theory is based upon the quasilinear properties of the SOAS to forced inputs, which permits the extension of quantitative linear feedback theory to the SOAS. A reasonable definition of optimum design is shown to be the minimization of the limit cycle frequency ω 0. The great advantages of the SOAS is its zero sensitivity to pure gain changes. However, quasilinearity and control of the limit cycle amplitude at the system output, impose additional constraints which partially or completely cancel this advantage, depending on the numerical values of the design parameters. By means of narrow-band filtering, an additional factor is introduced which permits trade-off between filter complexity and ω 0 minimization. It is shown that any SOAS design is inherently sensitive to external disturbances, and it is shown how to find the extreme disturbance for which quasilinearity holds. The theory permits optimum design to achieve specified bounds on system response to command inputs, despite given ranges of plant parameter uncertainty.